Purpose

The variational principle views a complex problem in an energy way, it gives good physical understanding of an iteration method, and the variational-based numerical methods always have a conservation scheme with a fast convergent rate. The purpose of this paper is to establish a variational principle for a fractal nano/microelectromechanical (N/MEMS) system.

Design/methodology/approach

This paper begins with an approximate variational principle in literature for the studied problem, and a genuine variational principle is obtained by the semi-inverse method.

Findings

The semi-inverse method is a good mathematical tool to the search for a genuine fractal variational formulation for the N/MEMS system.

Research limitations/implications

The established variational principle can be used for both analytical and numerical analyses of the N/MEMS systems, and it can be extended to some more complex cases.

Practical implications

The variational principle can be used for variational-based finite element methods and energy-based analytical methods.

Originality/value

The new and genuine variational principle is obtained. This paper discovers the missing piece of the puzzle for the establishment of a variational principle from governing equations for a complex problem by the semi-inverse method. The new variational theory opens a new direction in fractal MEMS systems.

中文翻译：

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分形纳米/微机电 (N/MEMS) 系统的变分原理

目的

变分原理以能量的方式看待复杂问题，它给出了迭代方法的良好物理理解，并且基于变分的数值方法总是具有收敛速度快的守恒格式。本文的目的是为分形纳米/微机电 (N/MEMS) 系统建立变分原理。

设计/方法/途径

本文针对所研究的问题，从文献中的近似变分原理入手，通过半逆的方法得到了真正的变分原理。

发现

半逆方法是一种很好的数学工具，可以为 N/MEMS 系统寻找真正的分形变分公式。

研究局限性/影响

建立的变分原理可用于 N/MEMS 系统的解析和数值分析，并且可以扩展到一些更复杂的情况。

实际影响

变分原理可用于基于变分的有限元方法和基于能量的分析方法。

原创性/价值

获得了新的真正的变分原理。本文找到了用半逆方法从控制方程中建立复杂问题变分原理的难题。新的变分理论开辟了分形 MEMS 系统的新方向。